32 problems

Let $F(z)\in\mathbb{C}[[z]]$ and let $\alpha\in\mathbb{C}$. The following conditions are equivalent:nn1. $F$ is Möbius-covariant of weight $\alpha$, namelyn … n2. For every nonnega…

Let $E_k$ denote the limit of the expected width of the largest block-palindrome factorization of a length-$n$ word over an alphabet of size $k$, whose existence is established for…

Let $\mathfrak g$ be a finite-dimensional Lie algebra, let $n$ be a positive integer, and let $x_1,\ldots,x_n\in\mathfrak g$. Define … For each $i$, let $\partial_{x_i}F_i$ be the…

Algebraicity criterion. The generalized double Laurent series is algebraic over $\mathbb{F}_q(t,u)$ if and only if: (a) for some positive integers $a$ and $b$,…

Let $F$ be an analytic map defined in an open neighborhood of $0\in{\mathbb C}^n$, and let $a(z)$ denote the formal series associated with $F$ through the formal differential opera…

Let $r(x)$ be a monic polynomial of degree $N+1>1$ with distinct roots, and let $\lambda(x)$ and $\mu(x)$ be the unique polynomials satisfying … with $\deg\mu(x)\leq N-1$ and…

Divergence criterion. The map $F(z)$ is polynomial if and only if $G(z)$ is polynomial. The source presents this as an algebraic formulation equivalent to the Jacobian conjecture,…

Twisted conjugacy growth series conjecture. The group $G$ is virtually abelian if and only if all its twisted conjugacy growth functions are rational. This is presented as a twiste…

Greedy 0-monoid conjecture. The formal power series $g(x)$ is a regular semi-upho function if and only if it admits an infinite greedy $0$-monoid series $\{M_k^0\}_{k\ge 1}$.

Catalan-polynomial structure conjecture. There exists a unique power series $P(x)$ with non-negative integer coefficients such that every $V_i(z)$ is the product of $C(z)$ and a po…

Let $S$ be an additively reduced and additively atomic semidomain, and let $cmathcal{A}_{+}(S)$ denote the set of additive atoms of $S$ and $S^{\times}$ its multiplicative unit gro…

For each $k\geq 1$, let $A_k(x):=\sum_{n\geq 0}a_{k,n}x^n$ be the modified formal power series introduced from the arrival recurrence, and let $A_k'(0)$ denote its derivative at ze…

Let $k$ be the coefficient field, let $d_1,d_2,d_3$ be nonnegative integers, and write $x=(x_1,\ldots,x_{d_1})$, $y=(y_1,\ldots,y_{d_2})$, and $z=(z_1,\ldots,z_{d_3})$. For a tuple…

Formal identity conjecture. One has

The Valqui–Guccione–Guccione criterion. Then

Let $G$ be a finitely presented group that is not virtually abelian, and let its conjugacy growth series be the formal power series whose coefficients are the values of the conjuga…

Sahi's coefficient-positivity conjecture. For every $n\geq 1$, the coefficient $c_n$ satisfies

Power-series extension conjecture. For every ideal $I$ of $R$,

Formal-power-series Broyden convergence conjecture. Broyden's method with $\nu_n=\frac{s_n}{s_n^t\cdot s_n}$ has locally Q-superlinear convergence. This is proposed as an adaptatio…

Lindahl's conjecture. The polynomial $f$ is linearizable if and only if $a_i=0$ for every $i\geq 2$ such that $p\nmid i$. This conjecture refines Lindahl's known examples of non-li…

Let $G$ be a finitely presented group that is not virtually abelian, and let a finite generating set mean any finite generating set of $G$. The general conjugacy-growth transcenden…

Let $R[[x,y]]$ be the ring of formal power series in $x$ and $y$ with coefficients in a … gi=1+alpha{i1}xui+alpha{i2}(xui)^2+alpha{i3}(xui)^3+cdots,qquad hi=1+beta{i1}yvi+beta{i2}(…

Rigidity Conjecture. If $a\circ b(X)\equiv X\pmod{X^{m+n+2}}$, then $a(X)=b(X)=X$. The paper introduces this conjecture after recording partial results on the length $2$ polydegree…

Let $m,n\in\mathbb N_+$ and let $a(X)\in{\mathbb C}[X]$ be a polynomial of degree at most $m+1$ such that $a(X)\equiv X\pmod{X^2}$. Let $a^{-1}(X)$ be its formal inverse for compos…

Let $m\in\mathbb N_+$ and let $a(X)\in{\mathbb C}[X]$ be a polynomial of degree at most $m+1$ such that $a(X)\equiv X\pmod{X^2}$. Let $a^{-1}(X)$ denote its formal inverse for comp…